Best Known (15, 15+18, s)-Nets in Base 2
(15, 15+18, 17)-Net over F2 — Constructive and digital
Digital (15, 33, 17)-net over F2, using
- net from sequence [i] based on digital (15, 16)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 15 and N(F) ≥ 17, using
(15, 15+18, 37)-Net over F2 — Upper bound on s (digital)
There is no digital (15, 33, 38)-net over F2, because
- 2 times m-reduction [i] would yield digital (15, 31, 38)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(231, 38, F2, 16) (dual of [38, 7, 17]-code), but
- residual code [i] would yield linear OA(215, 21, F2, 8) (dual of [21, 6, 9]-code), but
- residual code [i] would yield linear OA(27, 12, F2, 4) (dual of [12, 5, 5]-code), but
- residual code [i] would yield linear OA(215, 21, F2, 8) (dual of [21, 6, 9]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(231, 38, F2, 16) (dual of [38, 7, 17]-code), but
(15, 15+18, 39)-Net in Base 2 — Upper bound on s
There is no (15, 33, 40)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(233, 40, S2, 18), but
- the linear programming bound shows that M ≥ 667867 414528 / 75 > 233 [i]